Theorem rexex 3002

Description: Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)

Assertion

Ref	Expression
rexex	⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥𝜑)

Proof of Theorem rexex

Step	Hyp	Ref	Expression
1		df-rex 2918	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		exsimpr 1796	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥𝜑)
3	1, 2	sylbi 207	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 384  ∃wex 1704   ∈ wcel 1990  ∃wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-rex 2918

This theorem is referenced by:  reu3  3396  rmo2i  3527  dffo5  6376  nqerf  9752  supsrlem  9932  vdwmc2  15683  toprntopon  20729  isch3  28098  19.9d2rf  29318  volfiniune  30293  bnj594  30982  bnj1371  31097  bnj1374  31099  dfrdg4  32058  bj-0nelsngl  32959  bj-ccinftydisj  33100  poimirlem25  33434  mblfinlem3  33448  mblfinlem4  33449  clsk3nimkb  38338  stoweidlem57  40274